Triadic motifs and dyadic self-organization in the World Trade Network

In self-organizing networks, topology and dynamics coevolve in a continuous feedback, without exogenous driving. The World Trade Network (WTN) is one of the few empirically well documented examples of self-organizing networks: its topology strongly depends on the GDP of world countries, which in turn depends on the structure of trade. Therefore, understanding which are the key topological properties of the WTN that deviate from randomness provides direct empirical information about the structural effects of self-organization. Here, using an analytical pattern-detection method that we have recently proposed, we study the occurrence of triadic "motifs" (subgraphs of three vertices) in the WTN between 1950 and 2000. We find that, unlike other properties, motifs are not explained by only the in- and out-degree sequences. By contrast, they are completely explained if also the numbers of reciprocal edges are taken into account. This implies that the self-organization process underlying the evolution of the WTN is almost completely encoded into the dyadic structure, which strongly depends on reciprocity.Comment: 12 pages, 3 figures; Best Paper Award at the 6th International Conference on Self-Organizing Systems, Delft, The Netherlands, 15-16/03/201

Analytical maximum-likelihood method to detect patterns in real networks

In order to detect patterns in real networks, randomized graph ensembles that preserve only part of the topology of an observed network are systematically used as fundamental null models. However, their generation is still problematic. The existing approaches are either computationally demanding and beyond analytic control, or analytically accessible but highly approximate. Here we propose a solution to this long-standing problem by introducing an exact and fast method that allows to obtain expectation values and standard deviations of any topological property analytically, for any binary, weighted, directed or undirected network. Remarkably, the time required to obtain the expectation value of any property is as short as that required to compute the same property on the single original network. Our method reveals that the null behavior of various correlation properties is different from what previously believed, and highly sensitive to the particular network considered. Moreover, our approach shows that important structural properties (such as the modularity used in community detection problems) are currently based on incorrect expressions, and provides the exact quantities that should replace them.Comment: 26 pages, 10 figure

Economic networks in and out of equilibrium

Economic and financial networks play a crucial role in various important processes, including economic integration, globalization, and financial crises. Of particular interest is understanding whether the temporal evolution of a real economic network is in a (quasi-)stationary equilibrium, i.e. characterized by smooth structural changes rather than abrupt transitions. Smooth changes in quasi-equilibrium networks can be generally controlled for, and largely predicted, via an appropriate rescaling of structural quantities, while this is generally not possible for abrupt transitions in non-stationary networks. Here we study whether real economic networks are in or out of equilibrium by checking their consistency with quasi-equilibrium maximum-entropy ensembles of graphs. As illustrative examples, we consider the International Trade Network (ITN) and the Dutch Interbank Network (DIN). We show that, despite the globalization process, the ITN is an almost perfect example of quasi-equilibrium network, while the DIN is clearly an out-of-equilibrium network undergoing major structural changes and displaying non-stationary dynamics. Among the out-of-equilibrium properties of the DIN, we find striking early-warning signals of the interbank crisis of 2008.Comment: Preprint, accepted for SITIS 2013 (http://www.sitis-conf.org/). Final version to be published by IEEE Computer Society as conference proceeding

Enhanced Gravity Model of trade: reconciling macroeconomic and network models

The structure of the International Trade Network (ITN), whose nodes and links represent world countries and their trade relations respectively, affects key economic processes worldwide, including globalization, economic integration, industrial production, and the propagation of shocks and instabilities. Characterizing the ITN via a simple yet accurate model is an open problem. The traditional Gravity Model (GM) successfully reproduces the volume of trade between connected countries, using macroeconomic properties such as GDP, geographic distance, and possibly other factors. However, it predicts a network with complete or homogeneous topology, thus failing to reproduce the highly heterogeneous structure of the ITN. On the other hand, recent maximum-entropy network models successfully reproduce the complex topology of the ITN, but provide no information about trade volumes. Here we integrate these two currently incompatible approaches via the introduction of an Enhanced Gravity Model (EGM) of trade. The EGM is the simplest model combining the GM with the network approach within a maximum-entropy framework. Via a unified and principled mechanism that is transparent enough to be generalized to any economic network, the EGM provides a new econometric framework wherein trade probabilities and trade volumes can be separately controlled by any combination of dyadic and country-specific macroeconomic variables. The model successfully reproduces both the global topology and the local link weights of the ITN, parsimoniously reconciling the conflicting approaches. It also indicates that the probability that any two countries trade a certain volume should follow a geometric or exponential distribution with an additional point mass at zero volume

Exact maximum-likelihood method to detect patterns in real networks

In order to detect patterns in real networks, randomized graph ensembles that preserve only part of the topology of an observed network are systematically used as fundamental null models. However, their generation is still problematic. The existing approaches are either computationally demanding and beyond analytic control, or analytically accessible but highly approximate. Here we propose a solution to this long-standing problem by introducing an exact and fast method that allows to obtain expectation values and standard deviations of any topological property analytically, for any binary, weighted, directed or undirected network. Remarkably, the time required to obtain the expectation value of any property is as short as that required to compute the same property on the single original network. Our method reveals that the null behavior of various correlation properties is different from what previously believed, and highly sensitive to the particular network considered. Moreover, our approach shows that important structural properties (such as the modularity used in community detection problems) are currently based on incorrect expressions, and provides the exact quantities that should replace them.

Multiplexity versus correlation: the role of local constraints in real multiplexes

Several real-world systems can be represented as multi-layer complex networks, i.e. in terms of a superposition of various graphs, each related to a different mode of connection between nodes. Hence, the definition of proper mathematical quantities aiming at capturing the level of complexity of those systems is required. Various attempts have been made to measure the empirical dependencies between the layers of a multiplex, for both binary and weighted networks. In the simplest case, such dependencies are measured via correlation-based metrics: we show that this is equivalent to the use of completely homogeneous benchmarks specifying only global constraints, such as the total number of links in each layer. However, these approaches do not take into account the heterogeneity in the degree and strength distributions, which are instead a fundamental feature of real-world multiplexes. In this work, we compare the observed dependencies between layers with the expected values obtained from reference models that appropriately control for the observed heterogeneity in the degree and strength distributions. This leads to novel multiplexity measures that we test on different datasets, i.e. the International Trade Network (ITN) and the European Airport Network (EAN). Our findings confirm that the use of homogeneous benchmarks can lead to misleading results, and furthermore highlight the important role played by the distribution of hubs across layers.Comment: 32 pages, 6 figure

Binary versus non-binary information in real time series: empirical results and maximum-entropy matrix models

The dynamics of complex systems, from financial markets to the brain, can be monitored in terms of multiple time series of activity of the constituent units, such as stocks or neurons respectively. While the main focus of time series analysis is on the magnitude of temporal increments, a significant piece of information is encoded into the binary projection (i.e. the sign) of such increments. In this paper we provide further evidence of this by showing strong nonlinear relations between binary and non-binary properties of financial time series. These relations are a novel quantification of the fact that extreme price increments occur more often when most stocks move in the same direction. We then introduce an information-theoretic approach to the analysis of the binary signature of single and multiple time series. Through the definition of maximum-entropy ensembles of binary matrices and their mapping to spin models in statistical physics, we quantify the information encoded into the simplest binary properties of real time series and identify the most informative property given a set of measurements. Our formalism is able to accurately replicate, and mathematically characterize, the observed binary/non-binary relations. We also obtain a phase diagram allowing us to identify, based only on the instantaneous aggregate return of a set of multiple time series, a regime where the so-called `market mode' has an optimal interpretation in terms of collective (endogenous) effects, a regime where it is parsimoniously explained by pure noise, and a regime where it can be regarded as a combination of endogenous and exogenous factors. Our approach allows us to connect spin models, simple stochastic processes, and ensembles of time series inferred from partial information

Signs of universality in the structure of culture

Understanding the dynamics of opinions, preferences and of culture as whole requires more use of empirical data than has been done so far. It is clear that an important role in driving this dynamics is played by social influence, which is the essential ingredient of many quantitative models. Such models require that all traits are fixed when specifying the "initial cultural state". Typically, this initial state is randomly generated, from a uniform distribution over the set of possible combinations of traits. However, recent work has shown that the outcome of social influence dynamics strongly depends on the nature of the initial state. If the latter is sampled from empirical data instead of being generated in a uniformly random way, a higher level of cultural diversity is found after long-term dynamics, for the same level of propensity towards collective behavior in the short-term. Moreover, if the initial state is randomized by shuffling the empirical traits among people, the level of long-term cultural diversity is in-between those obtained for the empirical and uniformly random counterparts. The current study repeats the analysis for multiple empirical data sets, showing that the results are remarkably similar, although the matrix of correlations between cultural variables clearly differs across data sets. This points towards robust structural properties inherent in empirical cultural states, possibly due to universal laws governing the dynamics of culture in the real world. The results also suggest that this dynamics might be characterized by criticality and involve mechanisms beyond social influence.Comment: 16 pages, 7 figures; the same results as in version 3, but a shorter Introduction, Discussion and Conclusio

Community detection for correlation matrices

A challenging problem in the study of complex systems is that of resolving, without prior information, the emergent, mesoscopic organization determined by groups of units whose dynamical activity is more strongly correlated internally than with the rest of the system. The existing techniques to filter correlations are not explicitly oriented towards identifying such modules and can suffer from an unavoidable information loss. A promising alternative is that of employing community detection techniques developed in network theory. Unfortunately, this approach has focused predominantly on replacing network data with correlation matrices, a procedure that tends to be intrinsically biased due to its inconsistency with the null hypotheses underlying the existing algorithms. Here we introduce, via a consistent redefinition of null models based on random matrix theory, the appropriate correlation-based counterparts of the most popular community detection techniques. Our methods can filter out both unit-specific noise and system-wide dependencies, and the resulting communities are internally correlated and mutually anti-correlated. We also implement multiresolution and multifrequency approaches revealing hierarchically nested sub-communities with `hard' cores and `soft' peripheries. We apply our techniques to several financial time series and identify mesoscopic groups of stocks which are irreducible to a standard, sectorial taxonomy, detect `soft stocks' that alternate between communities, and discuss implications for portfolio optimization and risk management.Comment: Final version, accepted for publication on PR